The governors of all three states say they are considering legal action to block the president’s immigration directive to shield upward of 4 million illegal immigrants from deportation. The three governors – Mike Pence of Indiana, Scott Walker of Wisconsin, and Rick Perry of Texas – are all Republicans and all potential presidential candidates.

Meanwhile, Maricopa County Sheriff Joe Arpaio – who has clashed with the Obama administration in the past over immigration – filed a lawsuit Thursday to stop Mr. Obama's move. In a statement, he said the action would result in the "increased release of criminal aliens back onto streets of Maricopa County, Arizona, and the rest of the nation."

For his part, Governor Perry says illegal immigration costs his state $12 million a month just to secure the border. Suing the president is “probably a very real possibility,” he said during a panel discussion at the Republican Governors Association meeting in Boca Raton, Fla., on Wednesday.

Republicans in Congress are outraged by the president’s unilateral move to grant what they call temporary “amnesty” to millions of undocumented workers. They see it as another example of executive overreach. But a congressional lawsuit against the president would have dim prospects, because it’s hard to show that Congress or members of Congress are being harmed by the immigration directive.

“Everybody I’ve talked to says we don’t have standing,” Senator McCain told a group of reporters on Thursday.

Indeed, two law firms have backed out of commitments to help the US House in its attempt to sue the president over the Affordable Care Act. On Tuesday, the House engaged its third outside counsel: Jonathan Turley, an expert on constitutional law at George Washington University.

Courts traditionally prefer to let the White House and Congress fight over differences and not get involved themselves. Elections, they reason, are the ultimate judge of such disputes, which courts see as political.

But Senator McCain points to a Supreme Court ruling in June, in which the justices unanimously ruled against Mr. Obama and his recess appointments to the National Labor Relations Board (NLRB) and the Consumer Financial Protection Board.

The complaint was brought by an outsider, businessman Noel Canning, who argued that his business was adversely affected by the NLRB and its new appointees – who he claimed were invalidly appointed during a certain kind of Senate recess.